% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/Fundamental_Theorem_Algebra/prob_251__5369930_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:28:33.086

% Could-be-implicit typings (3)
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (18)
thf(sy_c_Complex_Ocomplex_ORe, type,
    re : complex > real).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex, type,
    abs_abs_complex : complex > complex).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal, type,
    abs_abs_real : real > real).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_real : real).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_v_r, type,
    r : real).
thf(sy_v_s, type,
    s : nat > complex).

% Relevant facts (247)
thf(fact_0_r, axiom,
    ((![N : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N)) @ r)))). % r
thf(fact_1_abs__1, axiom,
    (((abs_abs_real @ one_one_real) = one_one_real))). % abs_1
thf(fact_2_abs__1, axiom,
    (((abs_abs_complex @ one_one_complex) = one_one_complex))). % abs_1
thf(fact_3_abs__add__abs, axiom,
    ((![A : real, B : real]: ((abs_abs_real @ (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B))) = (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)))))). % abs_add_abs
thf(fact_4_add__le__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_left
thf(fact_5_add__le__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_left
thf(fact_6_add__le__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_right
thf(fact_7_add__le__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_right
thf(fact_8_rp, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % rp
thf(fact_9_abs__triangle__ineq, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (abs_abs_real @ (plus_plus_real @ A @ B)) @ (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)))))). % abs_triangle_ineq
thf(fact_10_abs__abs, axiom,
    ((![A : complex]: ((abs_abs_complex @ (abs_abs_complex @ A)) = (abs_abs_complex @ A))))). % abs_abs
thf(fact_11_abs__abs, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_abs
thf(fact_12_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_13_add__left__cancel, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_14_add__left__cancel, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_15_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_16_add__right__cancel, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_17_add__right__cancel, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_18_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_19_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_20_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_21_le__zero__eq, axiom,
    ((![N2 : nat]: ((ord_less_eq_nat @ N2 @ zero_zero_nat) = (N2 = zero_zero_nat))))). % le_zero_eq
thf(fact_22_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y)) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_23_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_24_add__cancel__right__right, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ A @ B)) = (B = zero_zero_real))))). % add_cancel_right_right
thf(fact_25_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_26_add__cancel__right__right, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ A @ B)) = (B = zero_zero_complex))))). % add_cancel_right_right
thf(fact_27_add__cancel__right__left, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ B @ A)) = (B = zero_zero_real))))). % add_cancel_right_left
thf(fact_28_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_29_add__cancel__right__left, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ B @ A)) = (B = zero_zero_complex))))). % add_cancel_right_left
thf(fact_30_add__cancel__left__right, axiom,
    ((![A : real, B : real]: (((plus_plus_real @ A @ B) = A) = (B = zero_zero_real))))). % add_cancel_left_right
thf(fact_31_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_32_add__cancel__left__right, axiom,
    ((![A : complex, B : complex]: (((plus_plus_complex @ A @ B) = A) = (B = zero_zero_complex))))). % add_cancel_left_right
thf(fact_33_add__cancel__left__left, axiom,
    ((![B : real, A : real]: (((plus_plus_real @ B @ A) = A) = (B = zero_zero_real))))). % add_cancel_left_left
thf(fact_34_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_35_add__cancel__left__left, axiom,
    ((![B : complex, A : complex]: (((plus_plus_complex @ B @ A) = A) = (B = zero_zero_complex))))). % add_cancel_left_left
thf(fact_36_double__zero__sym, axiom,
    ((![A : real]: ((zero_zero_real = (plus_plus_real @ A @ A)) = (A = zero_zero_real))))). % double_zero_sym
thf(fact_37_double__zero, axiom,
    ((![A : real]: (((plus_plus_real @ A @ A) = zero_zero_real) = (A = zero_zero_real))))). % double_zero
thf(fact_38_add_Oright__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.right_neutral
thf(fact_39_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_40_add_Oright__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.right_neutral
thf(fact_41_add_Oleft__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.left_neutral
thf(fact_42_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_43_add_Oleft__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.left_neutral
thf(fact_44_abs__zero, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_zero
thf(fact_45_abs__eq__0, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0
thf(fact_46_abs__0__eq, axiom,
    ((![A : real]: ((zero_zero_real = (abs_abs_real @ A)) = (A = zero_zero_real))))). % abs_0_eq
thf(fact_47_abs__0, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_0
thf(fact_48_abs__0, axiom,
    (((abs_abs_complex @ zero_zero_complex) = zero_zero_complex))). % abs_0
thf(fact_49_zero__le__double__add__iff__zero__le__single__add, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ (plus_plus_real @ A @ A)) = (ord_less_eq_real @ zero_zero_real @ A))))). % zero_le_double_add_iff_zero_le_single_add
thf(fact_50_double__add__le__zero__iff__single__add__le__zero, axiom,
    ((![A : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ A) @ zero_zero_real) = (ord_less_eq_real @ A @ zero_zero_real))))). % double_add_le_zero_iff_single_add_le_zero
thf(fact_51_le__add__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ (plus_plus_real @ B @ A)) = (ord_less_eq_real @ zero_zero_real @ B))))). % le_add_same_cancel2
thf(fact_52_le__add__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ (plus_plus_nat @ B @ A)) = (ord_less_eq_nat @ zero_zero_nat @ B))))). % le_add_same_cancel2
thf(fact_53_le__add__same__cancel1, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ (plus_plus_real @ A @ B)) = (ord_less_eq_real @ zero_zero_real @ B))))). % le_add_same_cancel1
thf(fact_54_le__add__same__cancel1, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ (plus_plus_nat @ A @ B)) = (ord_less_eq_nat @ zero_zero_nat @ B))))). % le_add_same_cancel1
thf(fact_55_add__le__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ B) @ B) = (ord_less_eq_real @ A @ zero_zero_real))))). % add_le_same_cancel2
thf(fact_56_add__le__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ B) @ B) = (ord_less_eq_nat @ A @ zero_zero_nat))))). % add_le_same_cancel2
thf(fact_57_add__le__same__cancel1, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ (plus_plus_real @ B @ A) @ B) = (ord_less_eq_real @ A @ zero_zero_real))))). % add_le_same_cancel1
thf(fact_58_add__le__same__cancel1, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ B @ A) @ B) = (ord_less_eq_nat @ A @ zero_zero_nat))))). % add_le_same_cancel1
thf(fact_59_abs__le__zero__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ zero_zero_real) = (A = zero_zero_real))))). % abs_le_zero_iff
thf(fact_60_abs__le__self__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ A) = (ord_less_eq_real @ zero_zero_real @ A))))). % abs_le_self_iff
thf(fact_61_abs__of__nonneg, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_nonneg
thf(fact_62_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_63_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_64_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_65_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_66_add_Ogroup__left__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.group_left_neutral
thf(fact_67_add_Ogroup__left__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.group_left_neutral
thf(fact_68_add_Ocomm__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.comm_neutral
thf(fact_69_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_70_add_Ocomm__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.comm_neutral
thf(fact_71_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_72_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_73_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_74_zero__neq__one, axiom,
    ((~ ((zero_zero_real = one_one_real))))). % zero_neq_one
thf(fact_75_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_76_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_77_abs__eq__0__iff, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0_iff
thf(fact_78_abs__eq__0__iff, axiom,
    ((![A : complex]: (((abs_abs_complex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % abs_eq_0_iff
thf(fact_79_complex__mod__triangle__sub, axiom,
    ((![W : complex, Z : complex]: (ord_less_eq_real @ (real_V638595069omplex @ W) @ (plus_plus_real @ (real_V638595069omplex @ (plus_plus_complex @ W @ Z)) @ (real_V638595069omplex @ Z)))))). % complex_mod_triangle_sub
thf(fact_80_add__nonpos__eq__0__iff, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ zero_zero_real) => ((ord_less_eq_real @ Y @ zero_zero_real) => (((plus_plus_real @ X @ Y) = zero_zero_real) = (((X = zero_zero_real)) & ((Y = zero_zero_real))))))))). % add_nonpos_eq_0_iff
thf(fact_81_add__nonpos__eq__0__iff, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ zero_zero_nat) => ((ord_less_eq_nat @ Y @ zero_zero_nat) => (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))))). % add_nonpos_eq_0_iff
thf(fact_82_add__nonneg__eq__0__iff, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ zero_zero_real @ X) => ((ord_less_eq_real @ zero_zero_real @ Y) => (((plus_plus_real @ X @ Y) = zero_zero_real) = (((X = zero_zero_real)) & ((Y = zero_zero_real))))))))). % add_nonneg_eq_0_iff
thf(fact_83_add__nonneg__eq__0__iff, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ zero_zero_nat @ X) => ((ord_less_eq_nat @ zero_zero_nat @ Y) => (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))))). % add_nonneg_eq_0_iff
thf(fact_84_add__nonpos__nonpos, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ B @ zero_zero_real) => (ord_less_eq_real @ (plus_plus_real @ A @ B) @ zero_zero_real)))))). % add_nonpos_nonpos
thf(fact_85_add__nonpos__nonpos, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => ((ord_less_eq_nat @ B @ zero_zero_nat) => (ord_less_eq_nat @ (plus_plus_nat @ A @ B) @ zero_zero_nat)))))). % add_nonpos_nonpos
thf(fact_86_add__nonneg__nonneg, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => (ord_less_eq_real @ zero_zero_real @ (plus_plus_real @ A @ B))))))). % add_nonneg_nonneg
thf(fact_87_add__nonneg__nonneg, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_eq_nat @ zero_zero_nat @ (plus_plus_nat @ A @ B))))))). % add_nonneg_nonneg
thf(fact_88_add__increasing2, axiom,
    ((![C : real, B : real, A : real]: ((ord_less_eq_real @ zero_zero_real @ C) => ((ord_less_eq_real @ B @ A) => (ord_less_eq_real @ B @ (plus_plus_real @ A @ C))))))). % add_increasing2
thf(fact_89_add__increasing2, axiom,
    ((![C : nat, B : nat, A : nat]: ((ord_less_eq_nat @ zero_zero_nat @ C) => ((ord_less_eq_nat @ B @ A) => (ord_less_eq_nat @ B @ (plus_plus_nat @ A @ C))))))). % add_increasing2
thf(fact_90_add__decreasing2, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ C @ zero_zero_real) => ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ B)))))). % add_decreasing2
thf(fact_91_add__decreasing2, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ C @ zero_zero_nat) => ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ B)))))). % add_decreasing2
thf(fact_92_add__increasing, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ B @ (plus_plus_real @ A @ C))))))). % add_increasing
thf(fact_93_add__increasing, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ B @ (plus_plus_nat @ A @ C))))))). % add_increasing
thf(fact_94_add__decreasing, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ C @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ B)))))). % add_decreasing
thf(fact_95_add__decreasing, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ B)))))). % add_decreasing
thf(fact_96_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_real @ one_one_real @ zero_zero_real))))). % not_one_le_zero
thf(fact_97_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_nat @ one_one_nat @ zero_zero_nat))))). % not_one_le_zero
thf(fact_98_zero__le__one, axiom,
    ((ord_less_eq_real @ zero_zero_real @ one_one_real))). % zero_le_one
thf(fact_99_zero__le__one, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ one_one_nat))). % zero_le_one
thf(fact_100_abs__ge__zero, axiom,
    ((![A : real]: (ord_less_eq_real @ zero_zero_real @ (abs_abs_real @ A))))). % abs_ge_zero
thf(fact_101_dual__order_Oantisym, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_102_dual__order_Oantisym, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_103_dual__order_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_104_dual__order_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_105_dual__order_Otrans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_eq_real @ C @ A)))))). % dual_order.trans
thf(fact_106_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_107_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_eq_real @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : real, B3 : real]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_108_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_109_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_110_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_111_order__trans, axiom,
    ((![X : real, Y : real, Z : real]: ((ord_less_eq_real @ X @ Y) => ((ord_less_eq_real @ Y @ Z) => (ord_less_eq_real @ X @ Z)))))). % order_trans
thf(fact_112_order__trans, axiom,
    ((![X : nat, Y : nat, Z : nat]: ((ord_less_eq_nat @ X @ Y) => ((ord_less_eq_nat @ Y @ Z) => (ord_less_eq_nat @ X @ Z)))))). % order_trans
thf(fact_113_order__class_Oorder_Oantisym, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_114_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_115_ord__le__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((B = C) => (ord_less_eq_real @ A @ C)))))). % ord_le_eq_trans
thf(fact_116_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_117_ord__eq__le__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % ord_eq_le_trans
thf(fact_118_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_119_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_120_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_121_antisym__conv, axiom,
    ((![Y : real, X : real]: ((ord_less_eq_real @ Y @ X) => ((ord_less_eq_real @ X @ Y) = (X = Y)))))). % antisym_conv
thf(fact_122_antisym__conv, axiom,
    ((![Y : nat, X : nat]: ((ord_less_eq_nat @ Y @ X) => ((ord_less_eq_nat @ X @ Y) = (X = Y)))))). % antisym_conv
thf(fact_123_le__cases3, axiom,
    ((![X : real, Y : real, Z : real]: (((ord_less_eq_real @ X @ Y) => (~ ((ord_less_eq_real @ Y @ Z)))) => (((ord_less_eq_real @ Y @ X) => (~ ((ord_less_eq_real @ X @ Z)))) => (((ord_less_eq_real @ X @ Z) => (~ ((ord_less_eq_real @ Z @ Y)))) => (((ord_less_eq_real @ Z @ Y) => (~ ((ord_less_eq_real @ Y @ X)))) => (((ord_less_eq_real @ Y @ Z) => (~ ((ord_less_eq_real @ Z @ X)))) => (~ (((ord_less_eq_real @ Z @ X) => (~ ((ord_less_eq_real @ X @ Y)))))))))))))). % le_cases3
thf(fact_124_le__cases3, axiom,
    ((![X : nat, Y : nat, Z : nat]: (((ord_less_eq_nat @ X @ Y) => (~ ((ord_less_eq_nat @ Y @ Z)))) => (((ord_less_eq_nat @ Y @ X) => (~ ((ord_less_eq_nat @ X @ Z)))) => (((ord_less_eq_nat @ X @ Z) => (~ ((ord_less_eq_nat @ Z @ Y)))) => (((ord_less_eq_nat @ Z @ Y) => (~ ((ord_less_eq_nat @ Y @ X)))) => (((ord_less_eq_nat @ Y @ Z) => (~ ((ord_less_eq_nat @ Z @ X)))) => (~ (((ord_less_eq_nat @ Z @ X) => (~ ((ord_less_eq_nat @ X @ Y)))))))))))))). % le_cases3
thf(fact_125_order_Otrans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % order.trans
thf(fact_126_order_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_127_le__cases, axiom,
    ((![X : real, Y : real]: ((~ ((ord_less_eq_real @ X @ Y))) => (ord_less_eq_real @ Y @ X))))). % le_cases
thf(fact_128_le__cases, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_eq_nat @ X @ Y))) => (ord_less_eq_nat @ Y @ X))))). % le_cases
thf(fact_129_eq__refl, axiom,
    ((![X : real, Y : real]: ((X = Y) => (ord_less_eq_real @ X @ Y))))). % eq_refl
thf(fact_130_eq__refl, axiom,
    ((![X : nat, Y : nat]: ((X = Y) => (ord_less_eq_nat @ X @ Y))))). % eq_refl
thf(fact_131_linear, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ Y) | (ord_less_eq_real @ Y @ X))))). % linear
thf(fact_132_linear, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) | (ord_less_eq_nat @ Y @ X))))). % linear
thf(fact_133_antisym, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ Y) => ((ord_less_eq_real @ Y @ X) => (X = Y)))))). % antisym
thf(fact_134_antisym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) => ((ord_less_eq_nat @ Y @ X) => (X = Y)))))). % antisym
thf(fact_135_eq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((ord_less_eq_real @ Y3 @ X2)))))))). % eq_iff
thf(fact_136_eq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[X2 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X2 @ Y3)) & ((ord_less_eq_nat @ Y3 @ X2)))))))). % eq_iff
thf(fact_137_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_138_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_139_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_140_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_141_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_142_ord__eq__le__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_143_ord__eq__le__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_144_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_145_order__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_146_order__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_147_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_148_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_149_order__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_150_order__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_151_order__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_152_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_153_add__right__imp__eq, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_154_add__right__imp__eq, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_155_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_156_add__left__imp__eq, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_157_add__left__imp__eq, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_158_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_159_add_Oleft__commute, axiom,
    ((![B : real, A : real, C : real]: ((plus_plus_real @ B @ (plus_plus_real @ A @ C)) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % add.left_commute
thf(fact_160_add_Oleft__commute, axiom,
    ((![B : complex, A : complex, C : complex]: ((plus_plus_complex @ B @ (plus_plus_complex @ A @ C)) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % add.left_commute
thf(fact_161_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_162_add_Ocommute, axiom,
    ((plus_plus_real = (^[A2 : real]: (^[B2 : real]: (plus_plus_real @ B2 @ A2)))))). % add.commute
thf(fact_163_add_Ocommute, axiom,
    ((plus_plus_complex = (^[A2 : complex]: (^[B2 : complex]: (plus_plus_complex @ B2 @ A2)))))). % add.commute
thf(fact_164_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A2 : nat]: (^[B2 : nat]: (plus_plus_nat @ B2 @ A2)))))). % add.commute
thf(fact_165_add_Oright__cancel, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_166_add_Oright__cancel, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_167_add_Oleft__cancel, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_168_add_Oleft__cancel, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_169_add_Oassoc, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % add.assoc
thf(fact_170_add_Oassoc, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % add.assoc
thf(fact_171_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_172_group__cancel_Oadd2, axiom,
    ((![B4 : real, K : real, B : real, A : real]: ((B4 = (plus_plus_real @ K @ B)) => ((plus_plus_real @ A @ B4) = (plus_plus_real @ K @ (plus_plus_real @ A @ B))))))). % group_cancel.add2
thf(fact_173_group__cancel_Oadd2, axiom,
    ((![B4 : complex, K : complex, B : complex, A : complex]: ((B4 = (plus_plus_complex @ K @ B)) => ((plus_plus_complex @ A @ B4) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add2
thf(fact_174_group__cancel_Oadd2, axiom,
    ((![B4 : nat, K : nat, B : nat, A : nat]: ((B4 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B4) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_175_group__cancel_Oadd1, axiom,
    ((![A4 : real, K : real, A : real, B : real]: ((A4 = (plus_plus_real @ K @ A)) => ((plus_plus_real @ A4 @ B) = (plus_plus_real @ K @ (plus_plus_real @ A @ B))))))). % group_cancel.add1
thf(fact_176_group__cancel_Oadd1, axiom,
    ((![A4 : complex, K : complex, A : complex, B : complex]: ((A4 = (plus_plus_complex @ K @ A)) => ((plus_plus_complex @ A4 @ B) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add1
thf(fact_177_group__cancel_Oadd1, axiom,
    ((![A4 : nat, K : nat, A : nat, B : nat]: ((A4 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A4 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_178_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((I = J) & (K = L)) => ((plus_plus_real @ I @ K) = (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_179_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_180_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_181_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_182_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_183_one__reorient, axiom,
    ((![X : real]: ((one_one_real = X) = (X = one_one_real))))). % one_reorient
thf(fact_184_one__reorient, axiom,
    ((![X : complex]: ((one_one_complex = X) = (X = one_one_complex))))). % one_reorient
thf(fact_185_add__le__imp__le__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) => (ord_less_eq_real @ A @ B))))). % add_le_imp_le_right
thf(fact_186_add__le__imp__le__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_eq_nat @ A @ B))))). % add_le_imp_le_right
thf(fact_187_add__le__imp__le__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) => (ord_less_eq_real @ A @ B))))). % add_le_imp_le_left
thf(fact_188_add__le__imp__le__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_eq_nat @ A @ B))))). % add_le_imp_le_left
thf(fact_189_le__iff__add, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (?[C2 : nat]: (B2 = (plus_plus_nat @ A2 @ C2)))))))). % le_iff_add
thf(fact_190_add__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)))))). % add_right_mono
thf(fact_191_add__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_right_mono
thf(fact_192_less__eqE, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => (~ ((![C3 : nat]: (~ ((B = (plus_plus_nat @ A @ C3))))))))))). % less_eqE
thf(fact_193_add__left__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)))))). % add_left_mono
thf(fact_194_add__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_left_mono
thf(fact_195_add__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ C @ D) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ D))))))). % add_mono
thf(fact_196_add__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_mono
thf(fact_197_add__mono__thms__linordered__semiring_I1_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_eq_real @ I @ J) & (ord_less_eq_real @ K @ L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(1)
thf(fact_198_add__mono__thms__linordered__semiring_I1_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_eq_nat @ I @ J) & (ord_less_eq_nat @ K @ L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(1)
thf(fact_199_add__mono__thms__linordered__semiring_I2_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((I = J) & (ord_less_eq_real @ K @ L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(2)
thf(fact_200_add__mono__thms__linordered__semiring_I2_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (ord_less_eq_nat @ K @ L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(2)
thf(fact_201_add__mono__thms__linordered__semiring_I3_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_eq_real @ I @ J) & (K = L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(3)
thf(fact_202_add__mono__thms__linordered__semiring_I3_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_eq_nat @ I @ J) & (K = L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(3)
thf(fact_203_abs__ge__self, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ (abs_abs_real @ A))))). % abs_ge_self
thf(fact_204_abs__le__D1, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ B) => (ord_less_eq_real @ A @ B))))). % abs_le_D1
thf(fact_205_abs__one, axiom,
    (((abs_abs_real @ one_one_real) = one_one_real))). % abs_one
thf(fact_206_norm__le__zero__iff, axiom,
    ((![X : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ X) @ zero_zero_real) = (X = zero_zero_complex))))). % norm_le_zero_iff
thf(fact_207_norm__le__zero__iff, axiom,
    ((![X : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X) @ zero_zero_real) = (X = zero_zero_real))))). % norm_le_zero_iff
thf(fact_208_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_209_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_210_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_211_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_212_norm__eq__zero, axiom,
    ((![X : complex]: (((real_V638595069omplex @ X) = zero_zero_real) = (X = zero_zero_complex))))). % norm_eq_zero
thf(fact_213_norm__eq__zero, axiom,
    ((![X : real]: (((real_V646646907m_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % norm_eq_zero
thf(fact_214__092_060open_062cmod_A_Is_A0_J_A_092_060le_062_Ar_092_060close_062, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (s @ zero_zero_nat)) @ r))). % \<open>cmod (s 0) \<le> r\<close>
thf(fact_215_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_216_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_217_abs__Re__le__cmod, axiom,
    ((![X : complex]: (ord_less_eq_real @ (abs_abs_real @ (re @ X)) @ (real_V638595069omplex @ X))))). % abs_Re_le_cmod
thf(fact_218_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_219_zero__complex_Osimps_I1_J, axiom,
    (((re @ zero_zero_complex) = zero_zero_real))). % zero_complex.simps(1)
thf(fact_220_one__complex_Osimps_I1_J, axiom,
    (((re @ one_one_complex) = one_one_real))). % one_complex.simps(1)
thf(fact_221_plus__complex_Osimps_I1_J, axiom,
    ((![X : complex, Y : complex]: ((re @ (plus_plus_complex @ X @ Y)) = (plus_plus_real @ (re @ X) @ (re @ Y)))))). % plus_complex.simps(1)
thf(fact_222_complex__Re__le__cmod, axiom,
    ((![X : complex]: (ord_less_eq_real @ (re @ X) @ (real_V638595069omplex @ X))))). % complex_Re_le_cmod
thf(fact_223_le0, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N2)))). % le0
thf(fact_224_add__is__0, axiom,
    ((![M : nat, N2 : nat]: (((plus_plus_nat @ M @ N2) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N2 = zero_zero_nat))))))). % add_is_0
thf(fact_225_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_226_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_227_nat__add__left__cancel__le, axiom,
    ((![K : nat, M : nat, N2 : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N2)) = (ord_less_eq_nat @ M @ N2))))). % nat_add_left_cancel_le
thf(fact_228_add__leE, axiom,
    ((![M : nat, K : nat, N2 : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ M @ K) @ N2) => (~ (((ord_less_eq_nat @ M @ N2) => (~ ((ord_less_eq_nat @ K @ N2)))))))))). % add_leE
thf(fact_229_le__add1, axiom,
    ((![N2 : nat, M : nat]: (ord_less_eq_nat @ N2 @ (plus_plus_nat @ N2 @ M))))). % le_add1
thf(fact_230_le__add2, axiom,
    ((![N2 : nat, M : nat]: (ord_less_eq_nat @ N2 @ (plus_plus_nat @ M @ N2))))). % le_add2
thf(fact_231_le__refl, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ N2 @ N2)))). % le_refl
thf(fact_232_add__leD1, axiom,
    ((![M : nat, K : nat, N2 : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ M @ K) @ N2) => (ord_less_eq_nat @ M @ N2))))). % add_leD1
thf(fact_233_add__leD2, axiom,
    ((![M : nat, K : nat, N2 : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ M @ K) @ N2) => (ord_less_eq_nat @ K @ N2))))). % add_leD2
thf(fact_234_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_235_eq__imp__le, axiom,
    ((![M : nat, N2 : nat]: ((M = N2) => (ord_less_eq_nat @ M @ N2))))). % eq_imp_le
thf(fact_236_le__Suc__ex, axiom,
    ((![K : nat, L : nat]: ((ord_less_eq_nat @ K @ L) => (?[N3 : nat]: (L = (plus_plus_nat @ K @ N3))))))). % le_Suc_ex
thf(fact_237_le__antisym, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) => ((ord_less_eq_nat @ N2 @ M) => (M = N2)))))). % le_antisym
thf(fact_238_add__le__mono, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ K @ L) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L))))))). % add_le_mono
thf(fact_239_add__le__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ K)))))). % add_le_mono1
thf(fact_240_nat__le__linear, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) | (ord_less_eq_nat @ N2 @ M))))). % nat_le_linear
thf(fact_241_trans__le__add1, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ I @ (plus_plus_nat @ J @ M)))))). % trans_le_add1
thf(fact_242_trans__le__add2, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ I @ (plus_plus_nat @ M @ J)))))). % trans_le_add2
thf(fact_243_nat__le__iff__add, axiom,
    ((ord_less_eq_nat = (^[M2 : nat]: (^[N4 : nat]: (?[K2 : nat]: (N4 = (plus_plus_nat @ M2 @ K2)))))))). % nat_le_iff_add
thf(fact_244_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y4 : nat]: ((P @ Y4) => (ord_less_eq_nat @ Y4 @ B))) => (?[X3 : nat]: ((P @ X3) & (![Y5 : nat]: ((P @ Y5) => (ord_less_eq_nat @ Y5 @ X3)))))))))). % Nat.ex_has_greatest_nat
thf(fact_245_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_246_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique

% Conjectures (1)
thf(conj_0, conjecture,
    ((![N3 : nat]: (ord_less_eq_real @ (abs_abs_real @ (re @ (s @ N3))) @ (plus_plus_real @ r @ one_one_real))))).
